The d5 break in the Sicilian

The Sicilian Dragon is my main weapon against e4 and my favourite opening. One crucial part of the Sicilian is to play d5 when you can unless it gives an apparent disadvantage. I always wondered what is the exact reason d5 is played. In the dragon, it's played to open up more lines on the queenside. That's as far as I know. I want to know why the d5 break is played in other systems. I thought it's played because it directly challenges white's centre and forces him to take otherwise he loses a pawn creating a 1:0 pawn influence for black on the centre. I've also heard it frees up the position for black which is why it's played in Scheveningen structures to open up the dark-squared bishop. And in e6 Sicilians to open up the light-squared bishop if you can't fianchetto. These were all guesses to me that seemed reasonable and I see the advantage. But what really confused me is the Prins variation. The main move is e5 and one of the main moves is d5 after. You don't take on e4 because it's defended by f3, and black pushes d4. PGN: 1. e4 c5 2. Nf3 d6 3. d4 cxd4 4. Nxd4 Nf6 5. f3 e5 6. Nb3 d5 7. Bg5 d4 If you don't understand what I am talking about. So the d4 move surprises me. What's the point of it. This entire question I just want to ask what's the point of the d5 break in the Sicilian? Is it a combination of what I said above? Is black threatening to play d4 in positions where white doesn't take and instead defends? Any experienced player please help me because this break is something I can never find the true purpose behind.

Tl;dr What is the main point of the d5 break in the Sicilian

The main point of ...d5 is to destroy central pawn e4.

To make it simple for you, the biggest reason for Black to play ...d5 is to activate the pieces. Activity of your pieces is in my book and those of GM the second most important thing in chess behind checkmating the opponent. Gambits are all about piece activity, even at a cost of material which says something about how important it is.

Less important is that you are ridding yourself of backward d6 pawn and attacking White's centre.

In most sicilian setups (not counting the alapin), black has a central majority, so it would make a lot of sense for black to push those pawns forward and occupy the centre with his majority, thus controlling more central squares and challenging white's e pawn! That's why it is considered good to play d5: You have an advantage in the centre and you are trying to prove it.

Everything you said is more or less correct. In the variation you reference (which seems to be the actual point of your question), black doesn't take on e4 because white would just recapture with the f-pawn, so black would not have achieved anything. Instead, black pushes his d-pawn again to grab space.

Your problem seems to be that you are expecting there to be only one purpose behind playing ...d5. Of course there are a few big reasons which dominate the main lines, but don't expect the same reasoning to work all the time. The reasons why a particular move is good or bad change drastically depending on the specifics of the position.

In this case, the move 5.f3 is a pretty slow move; in response, black seizes the initiative, first by playing 5...e5 to kick white's knight away, then 6...d5, grabbing space where he can. If 7.exd5 Nxd5 then black is certainly fine (more space, better placed pieces, and white's f3 move now looks pretty stupid), so instead white plays 7.Bg5, trying to prevent ...Nxd5; but now black continues 7...d4!, preventing 8.exd5 and also hampering white's development (the knight on b1 can no longer come to c3).

(Black could also continue, for example, 7...dxe4!? 8.Qxd8 Kxd8 9.fxe4. This is probably just fine for black, but it's not what I'd expect from somebody playing a 2...d6 Sicilian. The move 7...d4 seems more logical to me for the above reasons. Edit: in fact the engine points out 9.Nc3! in this variation, just giving up the pawn and preparing to castle queenside, which is obviously stronger than the immediate 9.fxe4.)